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--- radar:rcs [2018/05/28 10:41] – dianka
+++ radar:rcs [2026/04/28 18:23] (current) – mauro
@@ Line 1: / Line 1: @@
-TOC is ok for now!   --- //[[webmaster@localhost|DokuWiki Administrator]] 2018/04/24 16:17//
-====== Definition of radar cross section ======
+====== General definition ======
+Radar cross section is the measure of a target's ability to reflect radar signals in the direction of the radar receiver, i.e. it is a measure of the ratio of backscatter power per steradian (unit solid angle) in the direction of the radar (from the target) to the power density that is intercepted by the target.
+Radar cross section is a measure of power scattered in a given direction when a target is illuminated by an incident wave. RCS is normalized to the power density of the incident wave at the target so that it does not depend on the distance of the target from the illumination source. This removes the effect of the transmitter power level and distance to target when the illuminating wave decreases in intensity due to the inverse square spherical spreading. RCS is also normalized so that inverse square fall-off of scattered intensity due to the spherical spreading is not a factor so that we do not need to know the position of the receiver. RCS has been defined to characterize the target characteristics and not the effects of transmitter power, receiver sensivity, and the position of the transmitter and receiver distance. An other term for RCS is an echo area.
+The size and ability of a target to reflect radar energy can be summarized into a single term, σ, known as the radar cross-section, which has units of m². This unit shows, that the radar cross section is an area. If absolutely all of the incident radar energy on the target were reflected equally in all directions, then the radar cross section would be equal to the target's cross-sectional area as seen by the transmitter. In practice, some energy is absorbed and the reflected energy is not distributed equally in all directions. Therefore, the radar cross-section is quite difficult to estimate and is normally determined by measurement.
+The target radar cross sectional area depends of:
+• the airplane’s physical geometry and exterior features,
+• the direction of the illuminating radar,
+• the radar transmitters frequency,
+• the used material types.
+======Mathematical definition of radar cross section ======
 Electromagnetic waves, with any specified polarization, are normally diffracted or scattered in all  directions when incident on a target. These scattered waves are broken down into two parts. The first part is made of waves that have the same polarization as the receiving antenna. The other portion of the scattered waves will have a different polarization to which the receiving antenna does not respond.
@@ Line 30: / Line 47: @@
 The RCS defined by Eq.(4) is often referred to as either the monostatic RCS,the backscattered RCS, or simply target RCS.
 ====== RCS Prediction model =======
@@ Line 46: / Line 65: @@
 Radar cross section fluctuates as a function of radar aspect angle and frequency.For the purpose of illustration, isotropic point scatterers are considered.
 An isotropic scatterer is one that scatters incident waves equally in all directions.Consider the geometry shown in Fig.1. In this case, two unit isotropic scatterers are aligned and placed along the radar line of sight (zero aspect angle) at a far field range . The spacing between the two scatterers is 1 meter. The radar aspect angle is then changed from zero to 180 degrees, and the composite RCS of the two scatterers measured by the radar is computed.
-This composite RCS consists of the superposition of the two individual radar cross sections. At zero aspect angle, the composite RCS is $2m^{2}$ . Taking scatterer- 1 as a phase reference, when the aspect angle is varied, the composite RCS is modified by the phase that corresponds to the electrical spacing
+This composite RCS consists of the superposition of the two individual radar cross sections. At zero aspect angle, the composite RCS is $2m^{2}$ . Taking scatterer 1 as a phase reference, when the aspect angle is varied, the composite RCS is modified by the phase that corresponds to the electrical spacing
 between the two scatterers. For example, at aspect angle $10^{o}$ , the electrical spacing between the two scatterers is
@@ Line 185: / Line 204: @@
 The backscattered RCS for a perfectly conducting sphere is constant in the optical region. For this reason, radar designers typically use spheres of known cross sections to experimentally calibrate radar systems. For this purpose, spheres are flown attached to balloons. In order to obtain Doppler shift, spheres of known RCS are dropped out of an airplane and towed behind the airplane whose velocity is known to the radar
+===== Radar Cross section for non-spherical object =====
+The back-scattering radar cross-section area (RCS) of a non-spherical object, moving with respect to the radar
+should be considered variable over time due to the continuous variations of the target attitude.
+The RCS variations can be taken into account by treating the RCS as a stochastic process.
+The complete characterization of a stochastic process requires knowledge of the density of joint probabilities of each order.Since we do not have such data, we limit ourselves to consider a stochastic description of the phenomenon based on the moments of the first and second order (averages and correlation functions).
+\begin{equation}
+σ =\left\vert V^2(2) \right\vert k = \left\vert \textstyle \sum_{k=1}^N α_{i}\exp[j\frac{4π}{λ}δ_{i}] \right\vert^2
+\end{equation}
+<figure label>
+{{media:figure16.jpg?450 |}}
+<caption>Radar cross section for non-spherical object [(cite: Fundamentals of Radiolocation)] </caption>
+</figure>
+<figure label>
+{{ :media:figure23.jpg?450 |}}
+<caption> RCS of a 5 scatterers system: Polar diagram [(cite: Fundamentals of Radiolocation)]  </caption>
+</figure>
+<figure label>
+{{ :media:figure24.jpg?450 |}}
+<caption> RCS of a 5 scatterers system: Histogram [(cite: Fundamentals of Radiolocation)]  </caption>
+</figure>
+===== Example =====
+Let’s considering a simple case in which there are only two scatterers (N = 2). Let’s assume that the two scattering elements are identical and non-interacting, in the far field
+<figure label>
+{{ :media:figure25.jpg?200 |}}
+</figure>
+\begin{equation}
+δ = l*sin(θ)
+\end{equation}
+\begin{equation}
+∇φ =\frac{4π}{λ}lsin(θ)
+\end{equation}
+\begin{equation}
+∇φ =φ_{2}-φ_{1} =\frac{4πδ}{λ}
+\end{equation}
+\begin{equation}
+\left ( \frac{A}{2} \right )^2 = a^2cos^2(\frac{∇φ}{2}) =\frac{a^2}{2}(1+cos(∇φ)) =\frac{a^2}{2}[1+cos(4π \frac{l}{λ}sin(θ))]
+\end{equation}
+\begin{equation}
+σ_{tot} = 2σ_{1}[1+ cos(\frac{4πl}{λ}sin(θ))]
+\end{equation}
+<figure label>
+{{ :media:figure26.jpg?200 |}}
+</figure>
+<figure label>
+{{ :media:figure27.jpg?300 |}}
+</figure>
+<figure label>
+{{ :media:figure28.jpg?300 |}}
+</figure>
+This example has been taken from the lecture note.
+===== RCS of Corner Reflector =====
+Corner reflectors are used to generate a particularly strong radar echo from objects that would otherwise have only very low effective Radar cross section (RCS). A corner reflector consisting of two or three electrically conductive surfaces which are mounted crosswise (at an angle of exactly 90 degrees). Incoming electromagnetic waves are backscattered by multiple reflection accurately in that direction from which they come. Thus, even small objects with small RCS yield a sufficiently strong echo.
+The single areas of the corner reflector should be large compared to the radar wavelength. The larger a corner reflector is, the more energy is reflected. It can also be constructed in resonance with the radar wavelength. This will increase the echo signals again. To reduce losses caused by the earth's curvature the corner-reflectors are mounted as high as possible at the top of a mast at small boats. If too much wind load would disturb, then spherical or cylindrical disguised versions are used.
+If should be backscattered in a three-dimensional directions, the corner reflector must be constructed of three reflecting surfaces. If should be backscattered in three-dimensionally distributed directions, then the corner reflector must be made of three reflecting areas. It is then called trihedral corner reflector. It will take place three reflections then before the beam is reflected back in the original direction
+<figure label>
+{{ :media:figure29.jpg?450 |}}
+<caption> Corner-reflector [(cite: Fundamentals of Radiolocation)]  </caption>
+</figure>
+<figure label>
+{{ :media:figure30.jpg?450 |}}
+<caption> Image of Corner-reflector [(cite: Fundamentals of Radiolocation)]   </caption>
+</figure>
 =====  Ellipsoid =====
- An ellipsoid centered at (0,0,0) is shown in figure.9. It is defined by the following equation:
+ An ellipsoid centered at (0,0,0) is shown in fig.10. It is defined by the following equation:
 \begin{equation}
@@ Line 339: / Line 448: @@
 </figure>
+Fig. 4 describes the difference in the scattering of the radar signal in some types of shapes.
+<figure label>
+{{ :media:figure22.jpg?450 |}}
+<caption> Signal dispersion regarding target shape.</caption>
+</figure>
 ==== RCS of complex objects ====
@@ Line 358: / Line 473: @@
 This section presents the most commonly used RCS statistical models. Statistical models that apply to sea, land, and volume clutter, such as the Weibull and Log-normal distributions, will be discussed in a later chapter. The choice of a particular model depends heavily on the nature of the target under examination.
+Thus a convenient way to characterize the RCS of a generic target, is to consider it as a stochastic process, sometimes this can lead to even rough approximations. The most commonly used stochastic models are the
+four Swerling models.
+They qualify the variation over time of the RCS assigning to this a function of probability to the firstorder
+density and a trend of the correlation function which decreases rapidly (or slowly) with respect to the time constants (such as the dwell time td and the time Ts) scan.
 === Chi-Square of Degree 2m ===
@@ Line 371: / Line 491: @@
 where Γ(m) is the gamma function with argument m, and $σ_{av}$ is the average value. As the degree gets larger the distribution corresponds to constrained RCS values (narrow range of values). The limit $m \rightarrow ∞$ corresponds to a constant RCS target (steady-target case).
-===
+=== Swerling I and II (Chi-Square of Degree 2) ===
-Swerling I and II (Chi-Square of Degree 2) ===
-In Swerling I, the RCS samples measured by the radar are correlated throughout an entire scan, but are uncorrelated from scan to scan (slow fluctuation). In this case, the pdf is
+In Swerling I, the RCS samples measured by the radar are correlated throughout an entire scan, but are uncorrelated from scan to scan (slow fluctuation), in other word SwerlingI model assumes that the behavior of the RCS within the dwell time is strongly correlated In this case, the pdf is
 \begin{equation}
@@ Line 381: / Line 500: @@
 where σ_{av} denotes the average RCS overall target fluctuation. Swerling II target fluctuation is more rapid than Swerling I, but the measurements are pulse to pulse uncorrelated. This illustrated on fig.15. Swerling II RCS distribution is also defined by equation 34. Swerlings I and II apply to targets consisting of many independent fluctuating point scatterers of approximately equal physical dimensions.
-===
-Swerling III and IV (Chi-Square of Degree 4) ===
+=== Swerling III and IV (Chi-Square of Degree 4) ===
 Swerlings III and IV have the same pdf, and it is given by
@@ Line 390: / Line 509: @@
 \end{equation}
-The fluctuations in Swerling III are similar to Swerling I; while in Swerling IV they are similar to Swerling II fluctuations (see fig.15). Swerlings III and IV are more applicable to targets that can be represented by one dominant scatterer and many other small reflectors.
+The fluctuations in Swerling III are similar to Swerling I; while in Swerling IV they are similar to Swerling II fluctuations (see fig.26). Swerlings III and IV are more applicable to targets that can be represented by one dominant scatterer and many other small reflectors.
 <figure label>
@@ Line 396: / Line 515: @@
 <caption>Radar returns from targets with different Swerling fluctuations. Swerling V corresponds to a steady RCS target case. [(cite: Radar Systems Analysis and Design Using MATLAB)] </caption>
 </figure>
+<figure label>
+{{ :media:figure32.jpg?450 |}}
+<caption>Radar returns from targets with different Swerling fluctuations. Swerling V corresponds to a steady RCS target case. [(cite: Fundamentals of Radiolocation)] </caption>
+</figure>
+=== Application of Swerling models ===
+Considering that the target consists of several independent scatterers, using the central limit theorem, its complex echo has in phase and quadrature (I and Q) components Gaussian distributed with zero mean and variance equal and independent. Therefore, the amplitude
+\begin{equation}
+A = \sqrt{I^2-Q^2}
+\end{equation}
+It is distributed according to Rayleigh and its square that is proportional to the RCS is distributed exponentially. considering :
+\begin{equation}
+S = SNR =\frac{A^2}{2σ_{n}^2}
+\end{equation}
+In the fixed target hypothesis the detection probability was:
+\begin{equation}
+P_{D} =P_{D}(s) = 2e^{-s} \textstyle \int\limits_{\sqrt{-ln(P_{fa})}}^{∞} x.e^{-x^2}.I_{0}(2\sqrt{s}x) dx
+\end{equation}
+\begin{equation}
+P_{D} = \textstyle \int\limits_{0}^{∞} P_D(s)P(s) dx
+\end{equation}
+If we consider a SW1 or SW2 model:
+\begin{equation}
+P(s) = \frac{1}{S_{0}}e^{\frac{-s}{s_{0}}}U(s)
+\end{equation}
+ In the case of targets SW1 and SW2 there is an easier procedure the signals present on the components in phase and quadrature are given by the sum of the $v_{I}$ voltages (t) and $v_{Q} (t)$ associated with the echo produced by the target signal (assumed eg. SW2 type) with two $n_{I}$ signals (t) and $n_{Q}$ (t), representing an additive noise process assumed Gaussian
+\begin{equation}
+I(t) = v_{I}(t) + n_{I}(t)
+\end{equation}
+\begin{equation}
+Q(t) = v_{Q}(t) + n_{Q}(t)
+\end{equation}
+If the target follows a SW2 type model the statistics ofsignals $v_{I}$ and $V_{Q}$ is  also Gaussian and the envelope of the signals is Rayleigh
+\begin{equation}
+σ_{I}^2 = σ_{Q}^2 =σ_{s}^2 + σ_{n}^2 =σ_{n}^2(1+SNR)
+\end{equation}
+\begin{equation}
+SNR =\frac{σ_{s}^2}{ σ_{n}^2}
+\end{equation}
+\begin{equation}
+v(t) = \sqrt{I^2(t) + Q^2(t)}
+\end{equation}
+\begin{equation}
+P(v) = \frac{v}{σ^2}\exp(\frac{-v^2}{2σ^2})U(v)
+\end{equation}
+\begin{equation}
+σ^2 = σ_{s}^2 + σ_{n}^2
+\end{equation}
+\begin{equation}
+P_D =\exp(\frac{-V_{T}^2}{2σ_{n}^2(1+SNR)})
+\end{equation}
+\begin{equation}
+P_D =\exp[\frac{ln(P_fa)}{1+SNR}]
+\end{equation}
+\begin{equation}
+\frac{ln(P_{fa})}{ln(P_{D})} = 1+SNR
+\end{equation}
+with
+\begin{equation}
+P_fa=exp(\frac{-V_{T}^2}{2σ_{n}^2})
+\end{equation}
+<figure label>
+{{ :media:figure33.jpg?450 |}}
+<caption>variation of the probability of detection versus the signal to noise ratio [(cite: Fundamentals of Radiolocation)]</caption>
+</figure>
+<figure label>
+{{ :media:figure36.png?450 |}}
+<caption>Improvement of the SNR versus probabilty of detection [(cite: Fundamentals of Radiolocation)]</caption>
+</figure>
+The received signal, less than a constant factor coming from the radar equation, is equal to
+\begin{equation}
+P(t) = I^2(t) + Q^2(t)
+\end{equation}
+By definition, the voltage of the envelope signal v (t) relative to the echo received is equal to
+\begin{equation}
+v(t) = \sqrt{P(t)} = sqrt{σ}
+\end{equation}
+unless of a multiplicative constant.
+\begin{equation}
+P_{v}(v) = P_{∑}(σ=v^2)\frac{1}{|dv/dσ} = P_{∑}(σ=v^2).2.v
+\end{equation}
+\begin{equation}
+ P_{∑}(σ) = P_{v}(v=\sqrt{σ})\frac{dv(σ)}{σ} = P_{v}\sqrt{σ}\frac{1}{2sqrt{σ}}
+\end{equation}
+For example the power density function of RCS and the one of voltage are given by the following equations :
+\begin{equation}
+P_{∑}(σ) = \frac{1}{σ_{0}}\exp(\frac{-σ}{σ_{0}})U(σ)
+\end{equation}
+\begin{equation}
+P_{v}(v) = \frac{2v}{σ_{0}}\exp(\frac{-v^2}{σ_{0}})U(v)
+\end{equation}
+Some RCS models are derived by approximating a generic target with a set of N scattering elements.The power of the echo signal, and thus the RCS is equal to:
+\begin{equation}
+y = x_{1}^2 + x_{1}^2 + ... + x_{n}^2
+\end{equation}
+where $x_{i}$ are Gaussian variables. If they have zero mean and the same variance, the probability density function of the RCS y has the expression
+\begin{equation}
+f_{y}(y) = \frac{y^{N/2-1}}{(σ_{n}\sqrt{2})^N Γ(N/2)}\exp(\frac{-y}{2σ_{n}^2})U(y)
+\end{equation}
+It is noted that if N is an even, N = 2m number:
+- you get an exponential density function if m = 1. (SW1 AND SW2)
+– If m = 2 the density function of SW3 and SW4 models is obtained
+The previous function can be extended to a parameter m any (non-integer)
+\begin{equation}
+f_{y}(y) = \frac{y^{m-1}}{(σ_{n}\sqrt{2})^2m Γ(m)}\exp{(\frac{-y}{2σ_{}^2})}U(y)
+\end{equation}
+For aircraft targets the typical values are 0.9 m<m<2, while for satellites or cylinders are of the order of 0.3 <m <2.
+There are also other models such as:
+– the one proposed by RICE where it is assumed to have an object in which is identifiable a main scatter surrounded by many small random scatterers.
+– the Log-Normal model, obtained by calculating the exponential of a Gaussian variable having a positive average value.
+=== Stealth technology ===
+With the term “stealth” (being stealthy, clandestine) it is referred the military technology that aims to make an airplane or missile (or nay other object) a "nearly invisible" to enemy radar or any other form of revelation (eg. thermal, ect.).
+Stealth technology (or LO for "low observability") is not a single technology. It is a combination of technologies that attempt to greatly reduce the distances at which a person or vehicle can be detected; in particular radar cross section reductions, but also acoustic, thermal, and other aspects.
+At the basis of the stealth ability of an aircraft there is the combination of the effects due to
+particular materials and an appropriate shape of the object.
+<figure label>
+{{ :media:figure17.jpg?450 |}}
+<caption>Stealth of a aircraft [(cite: Fundamentals of Radiolocation)] </caption>
+</figure>
+=== Radar cross-section (RCS) reductions ===
+Almost since the invention of radar, various methods have been tried to minimize detection. Rapid development of radar during World War II led to equally rapid development of numerous counter radar measures during the period; a notable example of this was the use of chaff. Modern methods include Radar jamming and deception.
+The term "stealth" in reference to reduced radar signature aircraft became popular during the late eighties when the Lockheed Martin F-117 stealth fighter became widely known. The first large scale (and public) use of the F-117 was during the Gulf War in 1991. However, F-117A stealth fighters were used for the first time in combat during Operation Just Cause, the United States invasion of Panama in 1989.[22] Increased awareness of stealth vehicles and the technologies behind them is prompting the development of means to detect stealth vehicles, such as passive radar arrays and low-frequency radars. Many countries nevertheless continue to develop low-RCS vehicles because they offer advantages in detection range reduction and amplify the effectiveness of on-board systems against active radar homing threats.
+Example of radar cross section reduction:
+ - The reflection echo radar by a B-2 bomber Stealth frontally disposed is equal to -40 dBm2 ie 10-4 m2.
+<figure label>
+{{ :media:figure18.jpg?450 |}}
+<caption>Northrop B-2 Spirit [(cite: Fundamentals of Radiolocation)] </caption>
+</figure>
+- The JSF (Joint Strike Fighter) is seen with a RCS (Radar Cross Section) slightly higher, about -30 dBm2.
+<figure label>
+{{ :media:figure19.jpg?450 |}}
+<caption>Joint strike figther [(cite: Fundamentals of Radiolocation)] </caption>
+</figure>
+<figure label>
+{{ :media:figure20.jpg?450 |}}
+<caption>Shaping to reduce the radar cross section [(cite: Fundamentals of Radiolocation)]</caption>
+</figure>
+<figure label>
+{{ :media:figure20.jpg?450 |}}
+<caption>Shaping to reduce the radar cross section [(cite: Fundamentals of Radiolocation)]</caption>
+</figure>
+<figure label>
+{{ :media:figure31.jpg?450 |}}
+<caption> Typical value of RCS [(cite: Fundamentals of Radiolocation)]</caption>
+</figure>
@@ Line 403: / Line 726: @@
 publisher : by Chapman & Hall/CRC
 published : 2000
+[(cite: Fundamentals of Radiolocation)]
+title : Fundamentals of Radiolocation
+author :  Mauro Leonardi
+publisher : Lecture note
+published : 2018